The order of increasing O.N. of S in `S_(8), S_(2) O_(8)^(-2), S_(2) O_(3)^(-2), S_(4) O_(6)^(-2)` is given below -

Options

A. `S_(8) lt S_(2) O_(8)^(-2) lt S_(2) O_(3)^(-2) lt S_(4) O_(6)^(-2)`
B. `S_(2) O_(8)^(-2) lt S_(4)O_(6)^(-2) lt S_(8)`
C. `S_(2) O_(8) ^(-2) lt S_(8) lt S_(4) O_(6)^(-2) lt S_(2) O_(3)`
D. `S_(8) lt S_(2) O_(3)^(_2) lt S_(4) O_(6) ^(-2) lt S_(2) O_(8)^(-2)`

Explanation

To determine the order of increasing oxidation number (O.N.) of sulfur in the compounds \( S_8, S_2O_8^{2-}, S_2O_3^{2-}, S_4O_6^{2-} \), we need to calculate the oxidation state of sulfur in each compound step by step.

### Step 1: Calculate O.N. of S in \( S_8 \)
- **Analysis**: \( S_8 \) is a molecular form of sulfur where sulfur atoms are bonded together in a crown-like structure.
- **Calculation**: In this molecular form, the oxidation state of sulfur is 0.

**O.N. of S in \( S_8 \) = 0**

### Step 2: Calculate O.N. of S in \( S_2O_8^{2-} \)
- **Analysis**: This compound contains sulfur and oxygen. The oxidation state of oxygen is typically -2, but we also have a peroxy linkage (O-O bond) here.
- **Calculation**:
- Let the oxidation state of sulfur be \( x \).
- There are 6 oxygen atoms with -2 O.N. and 2 oxygen atoms with -1 O.N. (due to the peroxy linkage).
- The equation becomes:
\[ 2x + 6(-2) + 2(-1) = -2 \]
\[ 2x - 12 - 2 = -2 \]
\[ 2x - 14 = -2 \implies 2x = 12 \implies x = 6 \]

**O.N. of S in \( S_2O_8^{2-} \) = +6**

### Step 3: Calculate O.N. of S in \( S_2O_3^{2-} \)
- **Analysis**: In this compound, all oxygen atoms are in the -2 oxidation state.
- **Calculation**:
- Let the oxidation state of sulfur be \( x \).
- The equation becomes:
\[ 2x + 3(-2) = -2 \]
\[ 2x - 6 = -2 \implies 2x = 4 \implies x = 2 \]

**O.N. of S in \( S_2O_3^{2-} \) = +2**

### Step 4: Calculate O.N. of S in \( S_4O_6^{2-} \)
- **Analysis**: Similar to the previous calculations, we will use the oxidation state of oxygen as -2.
- **Calculation**:
- Let the oxidation state of sulfur be \( x \).
- The equation becomes:
\[ 4x + 6(-2) = -2 \]
\[ 4x - 12 = -2 \implies 4x = 10 \implies x = 2.5 \]

**O.N. of S in \( S_4O_6^{2-} \) = +2.5**

### Summary of O.N. Values
- \( S_8 \): 0
- \( S_2O_8^{2-} \): +6
- \( S_2O_3^{2-} \): +2
- \( S_4O_6^{2-} \): +2.5

### Step 5: Order of Increasing O.N.
Now, we can arrange the oxidation numbers in increasing order:
1. \( S_8 \) (0)
2. \( S_2O_3^{2-} \) (+2)
3. \( S_4O_6^{2-} \) (+2.5)
4. \( S_2O_8^{2-} \) (+6)

### Final Answer
The order of increasing oxidation number of sulfur is:
\[ S_8 < S_2O_3^{2-} < S_4O_6^{2-} < S_2O_8^{2-} \]